Elastic instability such as the buckling of cellular materials plays a pivotal role in their analysis and design. Despite extensive research, the quantifi- cation of critical stresses leading to elastic instabi- lities remains challenging due to the inherent nonlinearities. We develop an analytical approach considering the spectral decomposition of the elasticity matrix of two-dimensional hexagonal lattice materials. The necessary and sufficient condition for the buckling is established through the zeros of the eigenvalues of the elasticity matrix. Through the analytical solution of the eigenvalues, the conditions involving equivalent elastic properties of the lattice were directly connected to the mathematical requirement of buckling. The equivalent elastic properties are expressed in closed form using geometric properties of the lattice and trigonometric functions of a non-dimensional axial force parameter. The axial force parameter was identified for four different stress cases, namely, compressive stress in the longitudinal and transverse directions separately and together and torsional stress. By solving the resulting nonlinear equations, we derive exact analytical expressions of critical eigenbuckling stresses for these four cases. Crucial parameter combinations leading to minimum buckling stresses are derived analytically. The exact closed-form analytical expressions derived in the paper can be used for quick engineering design calculations and benchmarking related experimental and numerical studies.

弹性不稳定性，例如多孔材料的屈曲，在其分析和设计中起着关键作用。尽管进行了广泛的研究，但由于固有的非线性，导致弹性不稳定的临界应力的量化仍然具有挑战性。我们开发了一种分析方法，考虑到二维六边形晶格材料的弹性矩阵的光谱分解。屈曲的充分必要条件是通过弹性矩阵的特征值的零点建立的。通过特征值的解析解，涉及晶格等效弹性特性的条件与屈曲的数学要求直接相关。等效弹性属性使用晶格的几何属性和无量纲轴力参数的三角函数以闭合形式表示。确定了四种不同应力情况下的轴向力参数，即纵向和横向方向上的压缩应力和扭转应力。通过求解由此产生的非线性方程，我们推导出这四种情况下临界特征屈曲应力的精确解析表达式。通过分析推导出导致最小屈曲应力的关键参数组合。论文中推导出的精确闭式解析表达式可用于快速工程设计计算和基准相关的实验和数值研究。确定了四种不同应力情况下的轴向力参数，即纵向和横向方向上的压缩应力和扭转应力。通过求解由此产生的非线性方程，我们推导出这四种情况下临界特征屈曲应力的精确解析表达式。通过分析推导出导致最小屈曲应力的关键参数组合。论文中推导出的精确闭式解析表达式可用于快速工程设计计算和基准相关的实验和数值研究。轴向力参数被确定为四种不同的应力情况，即纵向和横向方向上的压缩应力和扭转应力。通过求解由此产生的非线性方程，我们推导出这四种情况下临界特征屈曲应力的精确解析表达式。通过分析推导出导致最小屈曲应力的关键参数组合。论文中推导出的精确闭式解析表达式可用于快速工程设计计算和基准相关的实验和数值研究。我们推导出这四种情况下临界特征屈曲应力的精确解析表达式。通过分析推导出导致最小屈曲应力的关键参数组合。论文中推导出的精确闭式解析表达式可用于快速工程设计计算和基准相关的实验和数值研究。我们推导出这四种情况下临界特征屈曲应力的精确解析表达式。通过分析推导出导致最小屈曲应力的关键参数组合。论文中推导出的精确闭式解析表达式可用于快速工程设计计算和基准相关的实验和数值研究。